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 A027363 Generalizing the 27 lines on a cubic surface: number of lines on the generic hypersurface of degree 2n-1 in complex projective (n+1)-space. 0

%I

%S 1,27,2875,698005,305093061,210480374951,210776836330775,

%T 289139638632755625,520764738758073845321,1192221463356102320754899,

%U 3381929766320534635615064019,11643962664020516264785825991165

%N Generalizing the 27 lines on a cubic surface: number of lines on the generic hypersurface of degree 2n-1 in complex projective (n+1)-space.

%D Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, Sequences of enumerative geometry: congruences and asymptotics, arXiv math.NT/0610286.

%D Van der Waerden, see one of his `Zur algebraischen Geometrie' papers.

%F Let b(n, i)=i/(n-i+1) and g(n, k)=s[ k ](b(n, 1), b(n, 2), ..., b(n, n)), where s[ k ] is the k-th elementary symmetric function; a(n) = (2n-1)^2 * (2n-2)! * [ g(2n-2, n-1) - g(2n-2, n) ].

%F a_n is the coefficient of x^{n-1} in the polynomial (1-X) prod_{j=0...2n-3}(2n-3-j+jX). [Van der Waerden]

%t a[n_] := Coefficient[ (1-x)*Product[ 2n-1-j+j*x, {j, 0, 2n-1}], x, n]; Table[a[n], {n, 1, 12}] (* From Jean-François Alcover, Jan 23 2012, from second formula *)

%K nonn,nice

%O 1,2

%A Paolo Dominici (pl.dm(AT)libero.it), Oct 15 1997

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